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2017 East Central NA Regional Contest

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2017-10-28 19:00 UTC

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Problem FKeeping On Track

Acmar and Ibmar are at war! You are in charge of a rail
network that transports important supplies throughout the great
state of Acmar during this delicate time. The rail system is
made up of a set of rail lines which meet at various junction
points. While there is no limit to the number of rail lines
that can meet at a junction, the network is set up so that
there is only one path between any two junctions. You’ve tried
to argue for some redundancy in the system, i.e., extra rail
lines so that there are two or more paths connecting some
junctions, but it’s wartime and budgets are tight.

However, this may soon change as you’ve just been given some
terrible news from double agents working in Ibmar: within the
next month enemy spies plan to blow up one of the junctions!
Unfortunately, the exact junction is not known, but knowing
your enemy well you are certain that they will undoubtedly
strike the critical junction, specifically the
junction whose removal disconnects the most pairs of other
remaining junctions in the system. You don’t have much time to
act, so the most you can do is add one new line connecting two
currently unconnected junctions, thereby reducing the number of
disconnected pairs after the critical junction has been
destroyed. Your job is to determine how to make the number of
disconnected pairs as small as possible by adding in the best
possible rail line.

Input

Input starts with a line containing an integer $n$ ($2
\leq n \leq 10\, 000$) indicating the number of rail
lines in the system. Following that are $n$ lines of the form $i_1$$i_2$ indicating that a rail line
connects junctions $i_1$
and $i_2$. Junctions are
numbered consecutively starting at $0$. All rail lines are two-way and no
rail line appears more than once in the input. There is exactly
one path between any two junction points given in the
input.

Output

Display two values $n_1$ and $n_2$, where $n_1$ is the number of pairs of
junctions which will be disconnected when the enemy destroys
the critical junction, and $n_2$ is the number of pairs of
junctions still disconnected after you add in the best possible
rail line. There will never be more than one critical
junction.